Module description

This module is designed to train students to participate in mathematics research under the direction of Dave Smith in the field of partial differential equations. Students will learn some preliminary topics from Fourier analysis and Complex analysis, then learn the unified transform method. Students will be expected to demonstrate their mastery of the method by implementing it to solve some classical and new initial boundary value problems. Students will also learn how to use modern techniques for collaborative research, including git and team task tracking. Students will be prepared for a number of summer research projects with funding available.

Learning outcomes

1. Apply Cauchy’s integral theorem and Jordan’s lemma to deform contour integrals of holomorphic functions.
   • Define a contour integral and evaluate simple contour integrals.
   • Describe interpretation of Cauchy’s integral theorem & Jordan’s lemma as contour deformation theorems.
   • Decide applicability of Cauchy’s integral theorem to analysis of a contour integral.
   • Use Cauchy’s integral theorem to evaluate appropriate contour integrals.
   • Decide applicability of Jordan’s lemma to analysis of a contour integral.
   • Use Jordan’s lemma to evaluate appropriate contour integrals.
2. Apply the unified transform method (UTM) to solve the prototypical linear evolution equations in 1+1d on the finite interval.
   • Implement stage 1 of UTM: derivation of global relation and Ehrenpreis form.
   • Implement stage 2 of UTM: proof of unicity and solution representation.
   • Implement stage 3 of UTM: proof of existence of solution.
   • Describe and mitigate issues including locus of zeros of characteristic determinant, dependence of well posedness upon choice of boundary conditions.
3. Discuss the relative merits of various techniques for analysing linear evolution equations in 1+1d.
   • Describe classical Fourier methods, including drawbacks and successes.
   • Describe applicability of UTM.
4. Employ modern techniques for collaborative research.
   • Work as part of a team on a major project.
   • Use git distributed version control software to save individual contributions to a major project.
   • Use GitLab issues to assign tasks and track progress on a major project.

Assessment

The module is letter graded and awards for 2MC, which may be counted towards an MCS major.

A significant portion of the module grade (60%) is awarded for a final project, which will be undertaken collaboratively, in groups of approximately 3 students. Otherwise, the module grade depends on participation in preparatory reading exercises and short individual problems assignments, most of which are undertaken towards the beginning of the module.

Summer research

The Dean of Faculty and CIPE are offering funding for up to 4 students completing this module to participate in summer research in this field. It may be possible to support further student research with funding from another source. If you are interested in doing summer research in mathematics, this is a good way to secure funding.

What kind of research?

Over the past couple of years, I have supervised seven summer student research projects on a variety of topics. One of these projects resulted in a paper submitted to an academic journal. You might also be interested to learn more about my own research.